Page 1
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Portia Isaacson Bass • Frank M. Bass Bass Economics, 4 Glenmoor Court, Frisco, Texas 75034
School of Management, University of Texas, Dallas, P.O. Box 830688, Richardson, Texas 75083 [email protected] • [email protected]
This paper explores multiple-generation demand dynamics of “fast-tech” products, which we define
as durable technological products and technology-based services where repeat purchases are moti-
vated by user-perceived functionality increases that trigger generational transitions. Examples of
fast-tech products include: personal computers (PCs), DRAMs, printers and wireless telephone ser-
vices. In management of fast-tech products, special attention must be paid to the different needs of
adopters and repeaters, which may require different product, advertising and distribution-channel
strategies. We develop a model of multiple-generation product diffusion in which sales are con-
structed as the sum of adoption sales and repeat sales thus, for the first time, separately identifying
first-time purchases and repeat purchases. The model also identifies (1) the potential market for each
generation, (2) total systems in use (subscribers if a service market) by time period and (3) systems-
in-use (installed-base) mix by product/service generation for each time period. The model reduces to
the basic Bass model (1969) in the case of a single generation. We use two sets of empirical data
(eight DRAM generations and nine PC generations) to demonstrate that the model provides an excel-
lent fit to historical data. We also provide support for the Norton-Bass Model by fitting it to these
same data.
1. Introduction
Technological products come in generations. The time between generations may be long as was the
case for sailing ships and ships powered with steam engines, or generations may follow one another
in rapid-fire succession as is true of semiconductor chips and personal computers. In any case, gen-
erational progression is a fundamental trait of technology-based products and, in a broader sense, of
all products. Generational boundaries are of great interest to management because the transitions are
rife with opportunity as well as fraught with danger. It is therefore important to develop an under-
Page 2
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 2 of 29
standing of the demand dynamics and interrelationships between product generations. The advance-
ment of such understanding is the purpose of this paper.
How is a generation to be defined? Because the basic phenomena being analyzed are based on
market observations of underlying buyer behavior, we believe that the appropriate definition should
be based on evaluations of product functionality from the buyer’s perspective and not alone on engi-
neering characterizations of individual product features. We shall suggest a general definition and
apply specific criteria for the product categories that we study.
There are two groups of buyers of a product generation: (1) adopters, who are first-time buyers
of the product category, and (2) repeat buyers, who previously purchased earlier generation products.
For a model to characterize inter-generational dynamics of product diffusion, it must capture the
processes of adoption and repeat buying. Although other models such as Norton-Bass (1987) accu-
rately model generation sales resulting from the underlying adoption and repeat-purchase processes,
the model presented here, which we shall refer to as the BB-01 Generations model, is the first to ex-
plicitly model the internal structure of these processes, constructing the sales of a generation from its
constitute adoption and repeat-purchase components.
Because we are especially interested in the diffusion of technology products and services, which
have relatively short inter-generational times (e.g., a few years), our desire is to develop models that
will stand the test of many generations. The greatest number of generations that have previously
been modeled is four. To verify the model presented here as well as to provide support for the valid-
ity of the Norton-Bass model (1987) we use empirical data from eight DRAM generations (1974-
2000) and nine PC generations (1975-2000).
2. Literature Review and Earlier Models
The earliest model of the demand relationship between one generation of a product and a successor
generation is the Fisher-Pry model (1971). This model is a simple model of substitution over time of
the later generation for the earlier expressed as a share of sales for each of the generations. The model
is based on Pearle’s Law: “The fractional rate of fractional substitution of new for old is proportional
to the remaining amount of the old left to be substituted.” The model is based on the assumption of
constant proportionality k and is expressed as the differential equation [ ]( ) ( ) 1 ( )d s t k s t s tdt
= − ,
Page 3
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 3 of 29
where s(t) is fractional market share. The Fisher-Pry model has been shown to provide good fits to
share data for two successive generations of products. Blackman (1971) modified and Perterka
(1977) extended the Fisher-Pry model to allow share to be distributed among several generations.
The Fisher-Pry model as well as its modifications deal with share only and do not account for sales of
the generations, nor does it account for the components of sales such as first-time buyers and repeat-
ers.
The first model of sales of multiple generations of products is the Norton-Bass model as de-
scribed in Norton (1986) as well as Norton and Bass (1987, 1992). This model is an extension of the
Bass model (1969) of the diffusion of a single-generation product. It deals with sales of successive
generations of products in those cases where adopters continue buying the product at a constant rate
and buyers of earlier generations gravitate to later generations according to the Bass model cumula-
tive distribution. In this model, each generation may expand the market. Any incremental market
adopts the newest generation or is usurped by a later generation in processes governed by Bass model
cumulative distributions. The Norton-Bass model has been shown to provide very good fits to sales
data for each product generation for several product categories. It has also been shown to dominate
the Fisher-Pry model in fitting share data. The Norton-Bass model equations for three generations
are:
[ ]1 1 1 2( ) ( ) 1 ( )s t F t m F t= − , (1)
[ ] [ ]2 2 2 1 1 3( ) ( ) ( ) 1 ( )s t F t m F t m F t= + − and (2)
[ ]3 3 3 2 2 1 1( ) ( ) ( ) ( )s t F t m F t m F t m = + + , (3)
where Sg(t) is sales of generation g at time t, tg is the time since the introduction of generation g, mg
represents the incremental market potential for generation g and F(tg) is the cumulative adoption
function of the Bass model at time tg. In Bass (1969), F(t) is shown to be
( )
( )
1( )1
p q t
p q t
eF tq ep
− +
− +
−=
+
, (4)
where p and q are model parameters to be estimated when the model is fit to observations data. In the
Norton-Bass model mg is conceptualized as
Page 4
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 4 of 29
g g gm a M= , (5)
where Mg is the ultimate market potential measured as the number of ultimate applications, customers
or sockets for generation g if there are no succeeding generations and ag is the average buying rate
per application per unit of time. Neither of these generational factors is observed or identified quanti-
tatively, but mg is a model parameter and is estimated.
In the model, sales of generation 1 will be m1F(t1) until generation 2 arrives and then it will be
this quantity minus what generation 2 steals. Migration from earlier generations to the latest genera-
tion is assumed to depend on the adoption rate for the latest generation and is the product of this
adoption rate and the theoretical sales of the prior generation if the latest generation had not arrived.
In other words, the Bass model diffusion process governs first-time sales as well as repeats. There
are two components of first-time sales: (1) customers who entered the market potential pool only
when the latest generation became available and who purchased the latest generation and (2) those
customers who made a first–time purchase of the latest generation but were in the pool of possible
adopters of the prior generation. The latter component was usurped from the prior generation by the
latest or it could be said to be leapfrogging the prior generation. Similarly, either generation g re-
peaters consist of those who
(1) made an adoption purchase or a repeat purchase of the prior generation and made a purchase of
generation g or (2) made a purchase of a generation earlier than g-1 and were usurped by generation g
thus leapfrogging generation g-1. Therefore, the model involves the usurping of both adopters and of
repeaters.
In equations (1), (2) and (3) for the Norton-Bass model the F functions do not have generational
subscripts because Norton and Bass found that for several product categories the p and q parameters
for the estimated F functions were the same across generations. A more general version of the model,
which was explored by Norton (1986), designates generational subscripts for the F’s and thus has p’s
and q’s that vary by generation.
There have been numerous applications and several published papers applying the Norton-Bass
model or some variation of it. Mahajan, Sharma and Buzzell (1991) used a similar model to estimate
the extent of Kodak’s infringement of Polaroid’s instant photography patents. Islam and Mead
Page 5
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 5 of 29
(1997) discuss use of the Norton-Bass model in which the p’s and q’s vary over generations as first
described by Norton (1986). Johnson and Bhatia (1997) applied the Norton-Bass model to genera-
tions of mobile communications and report good results. Kim, Chang, and Shocker (2000) modified
the Norton-Bass model to deal with inter-category effects as well as generational dynamics in a study
of wireless telecommunications. Bayus, Kim and Shocker (2000) have provided an overview and
literature review of models that treat the dynamics of multi-product interactions including successive
product generations. Danaher, Hardie and Putsis (2001) have developed a model of a subscription
service for two generations of analog wireless telephone. The model posits the basic Bass model as
the process for adoptions of the first generation product prior to the entry of the second generation
and posits usurping of the remaining market potential of the first generation by the second generation
at a rate that is proportional to the basic Bass model adoption rate for the expanded market potential.
They mistakenly conclude that the Norton Bass model (1987) does not allow leapfrogging, however
as described above, this is not true. Mahajan and Muller (1996) have developed a model to describe
the systems of IBM mainframe computers in use based on an extension of the basic Bass model and
have developed normative guidelines for the introduction timing of a new generation based on the
model. Lehmann and Pae (2000) have used the framework of the Norton-Bass model in a cross-
sectional analysis of several product categories to study the effect of intergeneration time, the time
between the introductions of successive generations, on the adoption rate for the new generation.
3. A Multiple-Generations Model of Adoptions and Repeat Sales
Prior models of diffusion of durable technological products and technology-based services have four
significant shortcomings. First, they do not separately identify first-time purchases and repeat pur-
chases. Second, they do not use the potential market as an identifiable quantity, which would be
helpful in both estimating and forecasting. Third, they do not permit calculation of the installed base
(products/services in use) at each time period. Fourth, they do not provide a way to quantify adopters
in each time period by the product generation they own, which is the installed-base mix by product
generation. Installed-base mix is often required; for example, PC software companies must track the
PC installed base by generation to estimate the number of PCs capable of running their application.
In the case of a subscription service (e.g., wireless telephone), the installed base is the number of sub-
Page 6
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 6 of 29
scribers and the installed-base mix quantifies the number of subscribers by generation of service (e.g.,
analog cellular, digital cellular, PCS) at each time period. To address these four shortcomings, a mul-
tiple-generations model of adoption and repeat sales is required.
Shortcomings of prior models are apparent in the modeling of “fast-tech” products. We define
“fast-tech” products as the thousands of product categories that improve so rapidly that customers
want a newer version long before the older one wears out or ceases to be capable of fulfilling previ-
ously defined needs. Fast-tech products include durable technology products (e.g., PCs, DRAMs,
microprocessors, printers, video games, PDAs) and technology services (e.g., wireless voice, soft-
ware, Web-based services, networks). When customers see the possibilities offered by a newer prod-
uct, they redefine their needs in light of the new technology. Fast-tech products include virtually all
products and services that are largely based on digital technology because the rapid changes in the
foundation technologies (e.g., semiconductor equipment, semiconductors, microprocessors, hard-disk
drives, telecommunications, software) enable rapid changes in products of which they are compo-
nents.
In fast-tech product markets, repeat purchases are motivated by functionality increases that trig-
ger generational transitions. Even in the early generations of fast-tech products and services, repeat
purchases are a substantial component of sales. For some generations repeat purchases may be the
largest group of purchases during the critical launch phase of a new product. A new generation may
also cause market expansion so that customers who would not have purchased a prior generation
product will adopt the new generation because of some combination of lower price, greater function-
ality, and greater ease of use.
Management of fast-tech products requires that special attention be paid to the different needs of
adopters and repeaters. Repeaters may require different product, advertising and distribution-channel
strategies. For example, repeaters may prefer higher-functionality, higher-priced products while
adopters may prefer lower-functionality, lower-priced products. Repeaters usually require a high de-
gree of operational compatibility with their old products while adopters might be more receptive to
new capabilities that are incompatible with prior generations. Repeaters may be more self-sufficient
than adopters and prefer not to pay for the expense of a high-support distribution channel. Mass-
Page 7
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 7 of 29
media advertising may be required to make new potential adopters aware of a new generation prod-
uct, while repeaters may be reachable via more targeted media.
We will develop a general class of multiple-generation diffusion models in which we model re-
peat sales and adoption sales (first-time purchases) explicitly and identify the sources of repeating
from earlier generations as well as repeat timing. We also model the sources and timing of adoption
sales in terms of first-time purchases usurped from prior generations and those that come from market
expansion. This class of models is the first to decompose sales into repeat sales and adoption sales.
We will refer to the new model as the BB-01 Generations model (or just the BB-01) to distinguish it
from other models.
3.1. “Generations” and Other Definitions
We define a product generation, or more exactly a product-category generation, as the set of product
brands and models fitting the customer-perceived functionality characteristics of the generation. For
example, 16-megabit (Mb) DRAM is a generation of the product category DRAM chips, which in-
cludes many chip brands and models from various manufacturers. The generation that followed was
64-Mb DRAM. We further require that a new generation of a product category be one triggered and
identified by a functionality increase so great that adopters of all prior generations will eventually buy
the new generation (or a later generation, if usurped). For example, the adopters of prior generations
were among the first adopters of the seventh PC generation (Windows PCs).
The model we develop can be used for both (1) end-user products and (2) products that are com-
ponents of end-user products. By definition for end-user products, each adoption (first-time pur-
chase) is of exactly one product unit and the adopter makes no further purchases of the product gen-
eration that was adopted. Further, by definition, each adopter after having made the first purchase
makes a repeat purchase of exactly one product unit in each successive generation except that a gen-
eration may be skipped by a portion of would-be repeaters when they leapfrog a generation (are
usurped by a later generation).
When modeling end-user products, sales are naturally expressed in user-perceived units. When
modeling products that are components of end-user products, however, the BB-01 definition of gen-
eration dictates that care be taken to express sales in user-perceived product units, not manufacturer-
Page 8
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 8 of 29
perceived units. For example, DRAM chips are components of PCs, but chips are manufacturer-
perceived units. PC end-users desire that their PCs have a specific amount of DRAM expressed in
megabytes (e.g., 64 MB) but do not generally know nor care how many DRAM chips are required.
Like the basic Bass model (1969), in the BB-01 model the maximum number of potential adopt-
ers Mg should be thought of as potential applications or potential “sockets” for the product, not the
number of potential purchasing units (customers). For example, the buying entity for PCs may be a
household, but the number of sockets might be best defined as the number of family members. Vari-
ously, the buying entity for PCs might be an office manager, but the sockets would be the number of
employees. Extending this approach might define two sockets per person with one for a desktop
computer and one for a notebook or one socket for a computer at home and one for a computer at the
office. A single buying entity (e.g., person, business, household) might purchase more than one end-
user product. To model such purchases, the units of the potential market Mg should be defined such
that no multiple purchases exist.
3.2. Model Components
The major components of the model structure are adoption sales and repeat sales. The basic equation
is
( ) ( ) ( )g g gsales t adoptionSales t repeatSales t= + (6)
or ( ) ( ) ( )g g gs t a t r t= + , (7)
where sg(t) is sales of product generation g, t is time relative to generation 1 starting at t=1, ag(t) is
adoption sales (sales to first-time buyers) at time t, rg(t) is repeat sales (sales to buyers who have pur-
chased a previous generation) at t.
3.2.1. The Adoption Function
In general terms ag may be expressed as a function of basic components:
1( ) ( ) ( ) ( )g g g ga t aMf t au t au t−= − + , (8)
where aMfg(t) is adoption sales that would have occurred if there had been no usurped sales, aug(t) is
adoption sales that are usurped by generation g+1, and aug-1(t) is adoption sales usurped by genera-
Page 9
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 9 of 29
tion g from generation g-1. Adoption sales, then, depend on the adoption process and the process of
usurpation.
Adoption sales of generation g at time t that would have occurred if there had been no usurped
sales is modeled as
( ) ( )g g g gaMf t M f t= , (9)
where Mg is the incremental market potential that is created by generation g, t is the starting time of
generation 1 and tg = t - τg + 1, 1 tg ug with τg being the time t that generation g was introduced
and ug being the number of periods for which there are sales of generation g. f(t) is the adoption-time
distribution of the basic Bass model as derived in Bass (1969) to be
2( )
2
( )
( )
( )
1
p q t
p q t
p q epf tq ep
− +
− +
+
= +
. (10)
Notice that Mg has the same meaning as in the basic Bass model, i.e. the maximum number of adopt-
ers in the market potential. To accomplish this we used the adoption function f of the Bass model
(1969) rather than the cumulative adoption function F that would have led us into the same difficul-
ties as in the Norton-Bass model where mg (see equation (5)) is a composite quantity that cannot be
separated into its constituents to identify Mg. Using f enables the BB-01 model to track the flow of
buyers from adoption to repeat purchasing.
The general usurped adoption sales function that we employ is:
1 1( ) ( ) ( )g g g gau t aMf t f t+ += , (11)
where aug(t) is the number of adopters usurped by generation g+1, aMfg(t) is the number of adopters
who would have adopted generation g if generation g+1 had not appeared and fg+1(tg+1) is the Bass
model adoption distribution for generation g+1. Many different functions might be chosen to repre-
sent the timing of usurpation, but because the adoption distribution for generation g+1 will reflect the
attractiveness of the new generation, one plausible assumption is that the usurpation distribution will
be the same as the adoption distribution for the usurping generation. Substituting equation (11) in
equation (8) we have
Page 10
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 10 of 29
11 1( ) ( ) ( ) ( ) ( ) ( )g g g gg g g ga t aMf t aMf t f t aMf t f t−+ += − + . (12)
3.2.2. The Repeat Function
In general terms the repeat function rg(t) may be expressed as:
1( ) ( ) ( ) ( ) ( )g g gg g gr t rpool t t ru t ru tφ −= − + , (13)
where rpoolg(t) is the pool of buyers available to make a repeat purchase of generation g, φg(tg) is the
repeat timing distribution, rpoolg(t) φg(tg) is repeat sales that would have been made at tg if there had
been no usurps, rug(t) is repeat sales that are usurped from generation g, and rug+1(t) is sales that are
usurped by generation g+1. We have examined different definitions of the repeat pool and different
specifications for the repeat timing distribution.
3.2.2.1. Repeat Pool Definition
For the repeat pool we considered the following definitions: (1) rpoolCAg(t), which is cumulative
adopters who bought any prior generation through t-1 and (2) rpoolCSg(t), which is cumulative sales
of the prior generation through t-1. We have explored other definitions of the repeat pool such as
cumulative sales over all generations prior to generation g but discarded them because they involve
double counting of buyers in the repeat pool. Consider repeat-pool definition (2), 1
11
( ) ( )g
t
gi
rpoolCS t s i−
−=
= ∑ , which is cumulative sales of the prior generation through t-1 and includes
some, but not all, of the adopters and repeaters of generations prior to generation g-1. The appeal of
(2) is that it ties repeating for the current generation to the buyers of the prior generation, but its
weakness is that at any time t it excludes adopters from generations 1 through g-2 who have not yet
purchased generation g-1. This exclusion produces a repeat pool that is too small, especially early in
the product life cycle for later generations. We have experimented with this definition in fitting the
model to observations of several product categories. We found that in models of products of only a
few generations (e.g., 2-4) it was satisfactory. For a larger number of generations (e.g., 8-9), how-
ever, the effect was to cause repeat sales to appear too late in the product life cycle, resulting in a
longer expected time-to-repeat that does not compare well with experience. Our preferred repeat
pool definition is definition (1), 1 1
1 1
( ) ( )g
g t
jj i
rpoolCA t a i− −
= =
= ∑∑ , which is cumulative adopters who bought
any prior generation through t-1. This definition is broader than (2) in that it includes adopters of all
Page 11
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 11 of 29
prior generations and after several generations is larger so it models repeats earlier in the product life
cycle yielding an expected time-to-repeat that compares favorably with experience.
3.2.2.2. Repeat Timing Distribution
In considering the repeat timing distribution φ(tg), we have examined the following specifications: (1)
a Bass model adoption distribution hg(tg) with additional parameters ag and bg to be estimated for
each generation, (2) a Bass model adoption distribution h(tg) with parameters a and b the same for all
generations and (3) the same Bass model adoption distribution fg(tg) as used to model adoptions.
Each of these employs a Bass model distribution because repeat purchasers like first-time purchasers
are influenced by external factors (e.g., advertising and publicity) and internal factors (e.g., product
recommendations by prior purchasers). After considering each of these alternatives including fitting
a version of the model using each, we concluded that the advantages of simplicity and fewer parame-
ters argued strongly for (3), the same distribution as used for adoptions, in spite of the potential ad-
vantage of (1) and (2) in modeling repeat timing independent of adoption timing. The repeat pool is
thus given by
1 1
1 1
( ) ( ) ( )g g
g t
jj i
rpool t rpoolCA t a i− −
= =
= = ∑∑ . (14)
The repeat function can now be expressed as
1( ) ( ) ( ) ( ) ( )g g gg g gr t rpool t f t ru t ru t−= − + , (15)
which by using ( ) ( ) ( )g g g grpoolf t rpool t f t= (16)
to expand becomes 1( ) ( ) ( ) ( )g g ggr t rpoolf t ru t ru t−= − + . (17)
3.2.2.3. Usurpation of Repeat Purchases
Using equation (16) we specify the usurping function rug(t) as
1 1( ) ( ) ( )g g g gru t rpoolf t f t+ += , (18)
thus allowing repeat purchases to be usurped in the same manner as adoption purchases. Substituting
in equation (15) we have the repeat function rg(t) as
11 1( ) ( ) ( ) ( ) ( ) ( )g g g gg g g gr t rpoolf t rpoolf t f t rpoolf t f t−+ += − + . (19)
Page 12
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 12 of 29
3.2.3. Equation to Be Estimated
From equation (7), we have ( ) ( ) ( )g g gs t a t r t= + . The components may be expanded using equations
(8) and (15) to obtain
1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )g g g g gg g g gs t aMf t au t au t rpool t f t ru t ru t− −= − + + − + , (20)
which with (9), (11), (14), (16) and (18) expands to the equation to be estimated:
1 1
1 1
1 1
1 21 1
1 11 1 1 1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ),
g g g
g g g g
g t
g jj i
g gt t
j g g j g gj i j i
s t a t a i f t
a i f t f t a i f t f t− −
− −
= =
− −− −
+ += = = =
= +
− +
∑∑
∑∑ ∑∑ (21)
where ag(t) is defined by equation (12) expanded with equation (9):
1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( )g g g g g g g g g g g g g ga t M f t M f t f t M f t f t+ + − − −= − + . (22)
After estimation of pg and qg of fg and Mg, the following components of sales can be derived for each
generation: adoption purchases before and after usurps, repeat purchases before and after usurps,
sales usurped from adoption purchases and sales usurped from repeat purchases.
3.2.4. Calculation of Installed Base and Installed-Base Generational Mix
Installed base IB(t) is used here as cumulative adopters and at time t is given
by 1 1 1
( ) ( ) ( )G t G
g gg i g
IB t a i cumA t= = =
= =∑∑ ∑ , (23)
where G is the number of generations and ag(i) is the number of adopters whose first purchase was a
generation g product at time i. Equation (23) counts adopters by the generation product they first
purchased. There is another way of counting adopters that is especially useful: by the generation of
product they own at time t, which if an adopter has made a repeat purchase, will be a later one than
they originally purchased. Using equation (7), (23) can be rewritten
1 11 1 1 1 1 1 1 1
( ) ( ) ( ) ( ) ( ) ( )G t t t G t t G
g g g g g gg i i i g i i g
a i r i r i s i r i IB t+ += = = = = = = =
+ − = − =
∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ , (24)
where 11 1
( ) ( ) ( ) ( ) ( )t t
g g g g gi i
s i r i cumS t CumR t IB t+= =
− = − = ∑ ∑ . (25)
In (25) IBg(t) is the number of adopters or repeaters who have purchased a generation g product by
time t and have not yet purchased a later generation product; that is, at time t, IBg(t) adopters own a
Page 13
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 13 of 29
generation g product. At t, the installed-base mix is given by the series IB1(t), IB2(t) . . . IBG(t). In
equations (23), (24) and (25) the quantities ag(t), sg(t) and rg(t) are net of usurps so no additional con-
sideration of usurps need be given. If the available data are the number of subscribers by type of ser-
vice (e.g., wireless telephone subscribers by generation of service), an alternative equation to be es-
timated is (25) rewritten as
11 1
( ) ( ) ( )t t
g g gi i
IB t s i r i+= =
= − ∑ ∑ . (26)
4. Models Fit to Empirical Data
The following sections describe the fit of the Norton-Bass model and the BB-01 model to eight gen-
erations of DRAM data and nine generations of PC data. In each case, the joint estimates of the pa-
rameters of the system of simultaneous nonlinear equations were developed utilizing SAS software.
4.1. Eight Generations of DRAM Data
The data employed are unit sales of dynamic random-access memory (DRAM) chips for eight genera-
tions sold between 1974 and 2000. The data are from Dataquest, the leading provider of DRAM
shipment data. DRAM generations are commonly identified by the number of bits per chip. As
shown in Figure 1, the chip generations modeled here are 4 Kb, 16 Kb, 64 Kb, 256 Kb, 1Mb, 4 Mb,
16 Mb and 64 Mb (Kb and
Mb indicate Kilobits and
megabits, respectively,
where K is 1024 and M is
10242 = 1,048,576). Chip
generations of configurations
greater than 64 Mb were in
their infancy at the time of
this study and were not in-
cluded. It is remarkable that
Figure 1 is consistent with
the previously observed em-
FIGURE 1: Worldwide DRAM Sales for 8 Generations
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
1974 1979 1984 1989 1994 1999
87 Observations
4 Kb
16 Kb
64 Kb256
1 Mb
4 Mb
16 Mb64 Mb
Generations > 64 Mb omitted
Page 14
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 14 of 29
pirical generalization pattern (Bass 1995) for the rise and fall of generations and market expansion as
generations increase. The generalization previously observed for four or fewer generations is seen
here to apply to eight generations of DRAM.
4.1.1. Norton-Bass Model Fit to DRAM Data
Figure 2 shows the Norton-Bass model with same p and q fit to eight generations compared to the 87
observations. Fitting the model to eight generations requires that 10 parameters be estimated: mg for
each generation and p and q of F, the Bass cumulative distribution. Also in Figure 2 are the parame-
ter estimates, standard errors and t-ratios as well as R-squares for each generation, which indicate that
the model provides a good fit to each generation.
FIGURE 2. Norton-Bass Model Fit to DRAM Chip Sales
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
1974 1979 1984 1989 1994 1999
ObservationsFittedEndodgenous
VariableR-
SquareAdj R-Square
S1(t) 0.922 0.924
S2(t) 0.757 0.761
S3(t) 0.843 0.845
S4(t) 0.980 0.980
S5(t) 0.908 0.908
S6(t) 0.979 0.979
S7(t) 0.966 0.964
S8(t) 0.919 0.913
4 Kb
16 Kb
64 Kb256
1 Mb
4 Mb
16 Mb64 Mb
Param EstimateApprox Std
Errt Ratio
m1 1.60E+08 6.83E+06 23.47m2 2.61E+08 3.89E+07 6.73m3 3.21E+08 6.63E+07 4.85m4 4.74E+08 6.17E+07 7.69m5 4.21E+08 8.83E+07 4.77m6 7.50E+08 9.92E+07 7.56m7 6.92E+08 1.17E+08 5.93m8 8.85E+07 1.99E+08 0.44
p 0.0180 0.0020 9.18q 0.8802 0.0321 27.45
4.1.2. BB-01 Model Fit of DRAM Observations
DRAMs are components in many types of end-user products such as PCs, printers, copiers and tele-
communication switches. PCs consume most DRAM chips; for example, according to Dataquest, in
1995 PCs were two-thirds of the DRAM market. With respect to the characteristics described below
the remainder of the DRAM market is thought to be similar to the PC market.
DRAM manufacturers identify DRAM generations as the number of bits per chip expressed in
kilobits (Kb) or megabits (Mb), but end-users of the products in which DRAMs are parts typically
Page 15
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 15 of 29
specify only the amount of DRAM bits desired, not the chip configuration. Because the bit is the
most convenient common factor for all chip configurations, a product socket is best though of as for a
single bit when employing the BB-01 model to fit DRAM generations data.
When an end-user makes a first-purchase of end-use equipment (e.g., PC), the component
DRAM bits are first-time purchases (adoptions). End users upgrade DRAM by adding chips or by
replacing chips with a later generation. The bits in added chips are also adoptions. The bits in re-
placement chips are repeat purchases up to the number of bits being replaced; over that the bits are
adoptions. Similar rules apply when the end-use equipment is replaced: bits in the new equipment up
to the number of bits in the equipment being replaced are repeats; others are adoptions. End-users of
products using DRAMs tend to expand or upgrade DRAM with each new DRAM generation because
(1) applications grow in memory requirement and (2) DRAM prices drop from generation to genera-
tion: typically a generation g bit costs one-fourth of a generation g-1 bit.
One might wonder whether DRAM satisfies the BB-01 model definition of a generation. Do us-
ers of end-use equipment with DRAM components virtually always replace and expand their DRAM
bits with each new DRAM generation? Remember that PC users can replace their DRAM in two
ways: (1) by replacing or adding DRAM chips or (2) by replacing the PC. There were eight genera-
tions of DRAM chips between 1975 and 2000 and nine generations of PCs. Clearly, the generations
cycles of these two product categories are on the average about the same. The reason is simply that
the other main drivers of PC generations (e.g., microprocessors, graphics controller, disk drive elec-
tronics) are on the same cycle as DRAMs because like DRAMs they are semiconductor products.
Because DRAM is on about the same generation cycle as its end-use products, the BB-01 model
can be used to fit DRAM sales in a three-step process. First, DRAM chips are changed to bits by
multiplying sales for each generation by the number of bits per chip for the generation. Second, the
BB-01 model is fit to bits sold per generation. Finally, fitted bit sales, Mg for each generation and
model components (e.g., adoptions, repeats, usurps) as well as the parameter standard errors are di-
vided by generational bits per chip to convert back to chips. Figure 3 shows the result compared to
the 87 observations. The model closely replicates the empirical generalization of the rise and fall of
generations and market expansion as generations increase. The fit is better than that of the Norton-
Page 16
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 16 of 29
Bass model with R-squares for four of the eight generations greater than .99 and with t-ratios quite
high. The fit is also superior to the Norton-Bass model fit with generational p’s and q’s (not shown).
FIGURE 3. Bass-Bass Model Fit to DRAM Chip Sales
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
1974 1979 1984 1989 1994 1999
Endodgenous Variable
R-Square
Adj R-Sq
S1(t) 0.994 0.994S2(t) 0.981 0.979S3(t) 0.926 0.927S4(t) 0.898 0.896S5(t) 0.929 0.926S6(t) 0.992 0.992S7(t) 0.993 0.992S8(t) 0.997 0.997
ObservationsFitted
4 Kb
16 Kb
64 Kb256
1 Mb
4 Mb
16 Mb64 Mb
Param EstimateApprox Std
Errt Ratio
M1 3.07E+08 5.44E+06 56.45M2 1.32E+09 4.63E+07 28.55M3 2.45E+09 1.62E+08 15.17M4 2.88E+09 3.18E+08 9.08M5 5.05E+09 3.68E+08 13.70M6 7.24E+09 1.84E+08 39.40M7 8.59E+09 2.41E+08 35.66M8 2.87E+09 1.07E+08 26.91
p1 0.0034 0.0004 9.31p2 0.0017 0.0004 4.42p3 0.0004 0.0002 1.96p4 0.0020 0.0012 1.69p5 0.0086 0.0023 3.70p6 0.0053 0.0005 9.81p7 0.0025 0.0003 7.76p8 0.0025 0.0003 7.88
q1 1.1056 0.0248 44.59q2 0.8723 0.0376 23.23q3 1.2764 0.0929 13.74q4 1.0135 0.1135 8.93q5 0.6476 0.0535 12.10q6 0.7694 0.0202 38.16q7 0.9901 0.0275 36.04q8 1.4409 0.0367 39.21
The fit of the BB-01 model is not the sole criterion for judging its usefulness. Rather, the infor-
mation content that can be generated from the model should also be considered. For example, Figure
4 shows the totals for all generations of DRAM chip adoptions and separately of repeats while Figure
5 shows the layers of adoptions and repeat purchases. Remarkably, even after 27 years, adoptions are
still by far the largest portion of sales. Bits per chip have increased so rapidly, quadrupling each gen-
eration, that repeat purchasers have recently bought fewer chips each generation and still increased
the number of bits per end-use device (e.g., PC). The result has been that for DRAM manufacturers
to achieve chip sales growth they have had to rely on market expansion.
Page 17
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 17 of 29
Figure 4. DRAM Adoptions v. Repeats (Left)Figure 5. DRAM Adoptions v. Repeats (Stacked) (Right)
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
3.0E+09
3.5E+09
4.0E+09
1974 1979 1984 1989 1994 1999
AdoptionsRepeats
Generations after 64 Mb are Missing
0.0E+00
5.0E+08
1.0E+09
1.5E+09
2.0E+09
2.5E+09
3.0E+09
3.5E+09
1974 1979 1984 1989 1994 1999
Total Adoptions Total Repeats
Generations after 64 Mb are Missing
4.2. Nine Generations of Personal Computers
Sales data for PCs by generation suitable for the BB-01 model are not available. Data categorizing
PC sales by microprocessor bit-width (e.g, 8-bits, 16-bits, 32-bits) are sometimes used, but this defi-
nition of generations identifies only three for the entire 26-year history of PCs. We develop below
the required generational definitions capturing all primary user-perceived PC generations.
Although PC data by generation are not available, total sales by year (see Figure 6) are, albeit
not from a single source. We used publications and databases from Future Computing, StoreBoard,
Computer Industry Almanac, BIS, Computer Industry Forecasts and eTForecasts as well as over a
million articles to glean the data and supporting qualitative information. We used data for only the
U.S. market because worldwide data are not as often based on reliable measurements. We included
PCs sold to both consumers and businesses. We included all operating systems (e.g., Basic, CP/M,
DOS, Apple II, Macintosh, Windows, Unix) and all PC forms (e.g., desktop computers, notebook
computers, small servers) except handhelds. We used an all-inclusive approach because the various
segments are so intricately intertwined that they are not separable. For example, a business user is a
consumer at home so consumer and business PC diffusion are not separable. We also used estimates
of installed base (see Figure 7) although we considered these estimates less reliable.
Page 18
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 18 of 29
Figure 6. U.S. PC Unit Sales Figure 7. U.S. PC Installed Base
0.0E+00
2.0E+07
4.0E+07
6.0E+07
8.0E+07
1.0E+08
1.2E+08
1.4E+08
1.6E+08
1.8E+08
1975 1980 1985 1990 1995 2000
0.E+00
1.E+07
2.E+07
3.E+07
4.E+07
5.E+07
1975 1980 1985 1990 1995 2000
The Slicer model was used to decompose total sales into the constituent-generation sales shown
in Figures 8 and 9 with generations named by the primary trigger (see Table 1). The Slicer model
methodology has three key steps. First, user-perceived generations are identified by start and end
dates by doing a thorough review of the category history with the goal of identifying all product
changes that triggered massive repeat purchasing. We used more than a million articles about the PC
market to glean
Figure 8. U.S. PC Unit Sales by Generation (Left)Figure 9. U.S. PC Unit Sales by Generation (Stacked) (Right)
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
Kits
Application Software
IBM PCCompatibility
HardDrive 32 Bits
Windows
Multimedia
Internet
Manufactured
52 Observations
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
Kits
Application Software
IBM PCCompatibility
HardDrive 32 Bits
Windows
Multimedia
Internet
Manufactured
Page 19
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 19 of 29
the detailed generational history, a
summary of which is shown in Table
1. In the second step of the Slicer
methodology, installed-base esti-
mates are used to calculate plausible
upper and lower bounds on cumula-
tive adopters for each generation.
Estimates of the installed base ex-
ceed cumulative adopters by the
number of systems that have been
replaced but not discarded. More-
over, installed-base estimates are
subject to error of unknown magni-
tude; therefore, the installed-base
numbers imply uncertainty about the
magnitude and increase in cumula-
tive adopters. The Slicer algorithm
employs the installed-base trajectory
to define bounds that are used dy-
namically to limit the percentage change in cumulative adopters (cumulative Mg) from generation to
generation. Using the bounds defined in step 2 in the third and final step, the Slicer model requires
that
1
( ) ( )G
g gg
TotalSales t s t=
= ∑ , (27)
where TotalSales(t) is sum of sales of all generations of a product category (e.g., PCs), G is the num-
ber of generations, t is the starting time of generation 1 and tg = t - τg + 1, 1 tg ug with τg being the
time t that generation g was introduced and ug being the number of periods for which there are sales
of generation g. In equation (27), sales of generation g at time tg is specified by
TABLE 1. U.S. PC Generations (1975-2000)
Gen
era
tio
n
Main Trigger (Name)
Secondary Drivers
Start-End Years
Typical Configuration Price New Thing
1 KitsMicro-processors, MS Basic
1975-1978
1-2 MHz 8080, 4-8KB RAM, Cassette Tape, Basic
$500-$800
Desktop PC Kits, Basic
2Manufactured PCs
CP/M, Floppy
1977-1980
1-2 MHz Z80, 16-32KB DRAM, Floppy Disk, CP/M, Basic
$1,500-$2,500
Manufactured PCs, CP/M OS, Floppy, Games
3Application Software
Home Computers
1979-1983
2-5 MHz Z80, 32-64KB DRAM, Hard Disk, Floppy, CP/M, Basic
$100-$300, $2,000-$3,500
Modem, VisiCalc WP, DB
4IBM PC Compatibility
Complete PC Standard
1982-1987
5-10 MHz Intel 8088, 64-256KB DRAM, HD/Floppy, MS DOS
$2,500-$4,000
Complete PC Standard, Portable PC, B&W Monitor
5 Hard Drive
Mac, LaserJet, Desktop Publishing
1984-1989
6-12.5 MHz Intel 286, 256KB-1MB DRAM, 10-40MB HD, Floppy, MS DOS, Macintosh
$2,000-$4,000
Laptop, Color Monitor, LAN, Mouse
6 32-BitsLarger Addressable RAM
1987-1993
16-33 MHz Intel 386, 1-4MB DRAM, 80-250MB HD, Floppy, MS DOS, Macintosh
$2,000-$5,000
Laser Printer, Desktop Publishing, Notebook PC
7 Windows Notebooks1990-1996
25-50 MHz Intel 486, 4-8MB DRAM, 300-600MB HD, Floppy, Windows 3.0
$1,700-$4,500
Client/Server, Color Printer,
8 MultimediaUnder $1,000 PCs, CD ROM
1993-2000
60-150 MHz Intel 486-Pentium, 8-16MB DRAM, 800MB-6GB HD, Floppy, CD ROM, Windows 3.1-95-NT
$1,000-$4,000
Web Browser, CD ROM, Sound, Color Notebook
9 Internet
Low-Price PCs, Email, Web Content, 56K+ Modem
1997-
150MHz-1GHz Intel Pentium II-III, 16-128MB DRAM, 8-30GB HD, Floppy, CD ROM, Windows 98-2000, Linux
$500-$3,500
CD-R, DVD
Page 20
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 20 of 29
1
( ) ( )g
g g g g gj
s t M f t=
= ∑ , (28)
where Mg is the incremental market potential of generation g and fg is the adoption distribution of the
basic Bass model as specified in equation (10). The set of series sg(tg), 1 g G, we refer to as the
resulting slice of total sales. By using cumulative market potential 1
g
gj
M=∑ as the quantity that is mul-
tiplied by fg(tg), the model dictates that the adopters of all prior generations as well as the adopters of
the current generation will purchase the current generation according to the Bass adoption distribu-
tion.
The Slicer algorithm that we have developed estimates the underlying sales for each PC genera-
tion. The algorithm requires that the sum of the estimated generational sales for each time period
closely approximate the known total sales for each time period. To generate the data shown in Fig-
ures 8 and 9, a combination of random and learning algorithms requiring intensive computing (ten
PCs running in parallel over several days) were used. The plausibility of the final slice was judged
by comparing known sales of a lead product (e.g., Windows) when available, to the generation sales
estimated by the algorithm. For detailed information about the algorithm, see Bass and Bass (2001).
4.2.1. Norton-Bass Generations Model Fit to PC-Generations Data
Figure 10 shows the Norton-Bass model fit to nine generations of PC data compared to the 52 obser-
vations. Fitting the Norton-Bass model to nine generations requires that 11 parameters be estimated:
mg for each generation and
Page 21
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 21 of 29
FIGURE 10. Norton-Bass Model Fit to PC Generations
Param EstimateApprox Std
Errt Ratio
m1 1.34E+05 5.28E+03 25.40m2 2.65E+05 2.55E+04 10.37m3 4.43E+05 1.01E+05 4.37m4 4.80E+06 3.70E+05 12.98m5 1.82E+06 5.54E+05 3.29m6 2.41E+06 6.79E+05 3.55m7 3.92E+06 5.69E+05 6.89m8 9.89E+06 4.85E+05 20.38m9 2.21E+07 6.26E+05 35.31
p 0.1593 0.0087 18.35q 1.0436 0.0493 21.16
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
ObservationsFittedEndodgenous
VariableR-
SquareAdj R-Square
S1(t) 0.956 0.957S2(t) 0.901 0.903S3(t) 0.720 0.724S4(t) 0.900 0.901S5(t) 0.908 0.908S6(t) 0.922 0.921S7(t) 0.993 0.992S8(t) 0.990 0.990S9(t) 0.998 0.998
Kits
Application Software
IBM PCCompatibility
HardDrive 32 Bits
Windows
Multimedia
Internet
Manufactured
p and q of F, the Bass cumulative distribution. Figure 10 also shows the parameter estimates with
parameter standard errors and t-ratios as well as R-squares for each generation. It is clear that the
Norton-Bass model provides an excellent fit to the nine generations of PCs. The t-ratios for all pa-
rameter estimates are very high and the R-squares for the last three generations are greater than .99.
4.2.2. BB-01 Generations Model Fit to PC-Generations Data
Figure 11 shows the BB-01 model fit to nine generations of PCs compared to the 52 observations.
Fitting the BB-01 model to nine generations requires that 27 parameters be estimated: pg and qg of fg
and Mg for each generation. Also shown are the parameter estimates with standard errors and t-ratios
as well as generational R-squares. The fit of the model is extraordinarily strong with R-squares for
each generation of .999 or greater. The high R-squares from the BB-01 model fit to the Slicer output
are not surprising since Slicer and the BB-01 model are based, albeit with different implementations,
on the same key insight into “fast-tech” markets: each adopter purchases one product each genera-
tion. Both the Slicer output and the BB-01 model fit of it should be judged, not on the BB-01 R-
squares, but on the plausibility of
Page 22
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 22 of 29
FIGURE 11. Bass-Bass Model Fit to PC Generations
Param EstimateApprox Std Err
t Ratio
M1 3.41E+05 9.17E+02 372.3M2 6.74E+05 1.28E+03 526.2M3 1.36E+06 2.85E+03 477.8M4 1.01E+07 2.00E+04 505.0M5 1.13E+07 6.62E+04 170.6M6 8.20E+06 1.23E+05 66.7M7 1.41E+07 1.69E+05 83.1M8 5.47E+07 3.38E+05 161.8M9 8.78E+07 3.30E+05 266.4
p1 0.0504 0.0003 155.1p2 0.0136 0.0001 191.4p3 0.0050 0.0000 145.6p4 0.0402 0.0002 164.6p5 0.0333 0.0003 98.0p6 0.0457 0.0005 87.0p7 0.0386 0.0004 95.4p8 0.0198 0.0002 79.7p9 0.0273 0.0000 579.2
q1 0.9726 0.0043 228.4q2 1.4443 0.0024 611.8q3 2.1041 0.0032 664.0q4 1.4311 0.0035 404.8q5 1.1903 0.0050 240.0q6 1.0520 0.0061 173.3q7 0.8970 0.0051 175.7q8 0.9268 0.0043 218.0q9 0.8962 0.0012 762.5
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
ObservationsFitted
Kits
Application Software
IBM PCCompatibility
HardDrive 32 Bits
Windows
Multimedia
Internet
Manufactured
Endodgenous Variable
R-Square
Adj R-Square
S1(t) 0.999 0.999S2(t) 0.999 0.999S3(t) 0.999 0.999S4(t) 0.999 0.999S5(t) 0.999 0.999S6(t) 0.999 0.999S7(t) 0.999 0.999S8(t) 0.999 0.999S9(t) 0.999 0.999
the resulting decomposition into adoption and repeat purchases as well as other obtainable informa-
tion. We considered and found highly plausible each of the following: (1) the evolution of repeat
purchases and adoptions for each generation, (2) the trend in repeats as a percentage of sales, (3) the
expected time-to-repeat trend, (4) the installed base over time, (5) the evolution of the installed-base
mix and (6) the fit of the basic Bass model to BB-01 adopters.
FIGURE 12. Components of Sales of Personal Computer Generation Seven
-1,000,000
1,000,000
3,000,000
5,000,000
7,000,000
9,000,000
11,000,000
1988 1993 1998
Adoptions Usurped from Gen 7 by Gen 8 Repeats Usurped from
Gen 7 by Gen 8
-1,000,000
1,000,000
3,000,000
5,000,000
7,000,000
9,000,000
11,000,000
1988 1993 1998
Gen 7 Adoptions
Gen 7 Repeats
Repeats Usurped by Gen 7 From Gen 6
Adoptions Usurpedby Gen 7 From Gen 6
Figure 12 shows the constituents of generation seven sales. Since the market is substantially de-
veloped by generation seven, it is not surprising that repeat purchases are the largest component fol-
Page 23
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 23 of 29
lowed by adoptions. Repeat purchases, adoptions and usurps increase from the start of generation
seven while adoptions and repeats usurped from generation seven start with eight.
In Figure 13, adoptions and repeats for each generation are shown. The upper line bounding the
total area closely approximates total U.S. PC sales in Figure 6. The graph shows that repeat pur-
chases contribute substan-
tially to sales after the first
few generations. Figure 14
and Figure 15 make clear
that repeats are generally
increasing with time except
when the market has been
substantially expanded by
the appeal of a new genera-
tion. The first such market
expansion occurred in 1982
when adoptions surged following the introduction of the IBM PC. In 1989 repeats peaked at 71% of
sales after which repeat purchases decreased relative to adoptions each year because of a succession
of new technologies that expanded the market; namely, Windows and notebook PCs in generation
FIGURE 13. BB-01 Estimate of PC Sales with Adoptions and Repeat Sales Separately Identified
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
AdoptersRepeats
Kits
Application Software
IBM PCCompatibility
HardDrive
32 Bits
Windows
Multimedia
Internet
Manufactured
FIGURE 14. Repeats and Adoptions (Stacked) (Left) FIGURE 15. Repeats as Percentage of Sales (Right)
0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
40,000,000
45,000,000
1975 1980 1985 1990 1995 2000
Repeats (All Gens)Adoptions (All Gens)
0%
10%
20%
30%
40%
50%
60%
70%
80%
1975 1980 1985 1990 1995 2000
Repeats % of Total
Page 24
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 24 of 29
seven, multimedia in generation eight and the Internet and under low-price PCs in generation nine.
From a low of 50% in 1998, repeat purchases started gaining again on adoptions. The reversals as
well as the percentages shown in Figure 15 are similar to those reported in the PC industry.
Another benchmark used to judge the plausibility of the BB-01 result is typical time between
purchases, which has been reported at times for some PC market segments. Figure 16 compares re-
ported typical time between purchases to calculated expected time-to-repeat (ETR). We calculated
ETR by assuming that repeaters make repeat purchases of generation g in the same order that they
purchased the prior generation.
By simulating a first-in-first-out
(FIFO) queue of generation g-1
purchasers, we are able to calcu-
late at any time t the ETR for
those leaving the FIFO queue by
making a repeat purchase of gen-
eration g. The annual average
ETR is a weighted average over
all generations. Figure 16 shows
bands of reported typical times
between purchases for three time intervals as a background for average annual ETR, which together
show plausibility of the expected time to repeat series. The 1981-82 ETR dip was due to the intro-
duction of the IBM PC, which attracted repeat purchases earlier than they would have otherwise oc-
curred.
Figure 17 shows cumulative adopters calculated by the BB-01 model compared to industry esti-
mates of installed base. Figure 18 shows the installed-base mix by year calculated by the BB-01
model, which quantifies PCs of a given vintage that are still in use.
The trends in Figure 17 are similar, both increasing with comparable slopes over much of the
range. The BB-01 installed base is lower and lags industry estimates by a few of years. As previ-
ously explained, estimates of the installed base exceed cumulative adopters by the number of PCs
FIGURE 16. BB-01 PC Expected Time-to-Repeat v. Typical Reported Time between Purchases
0
1
2
3
4
5
1975 1980 1985 1990 1995 2000
Years
Typical Reported Time Between Purchases: 1975-1980
Typical Reported TimeBetween Purchases: 1981-1993
Typical Reported Time Between Purchases: 1994-2000
Page 25
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 25 of 29
that have been replaced but not discarded; that is, the BB-01 model counts only the purchase of new
PCs as adoptions with the passing on of a used PC not considered an adoption. The fact that replaced
PCs are kept a few years by the same or another owner accounts for much of the discrepancy between
the model and industry estimates. Another contributing factor is that industry estimates are very
rough, typically based on spotty penetration surveys measuring only one segment of the market (e.g.,
consumer). Industry measures of PC sales have been much more reliable than estimates of installed
base since product flow (regardless of customer segment) through most distribution channels has
been accurately measured most of the 26-year history of personal computers.
As a final plausibility indicator, we used the basic Bass model (1969) to fit the adoptions series
from the BB-01 model. We have long understood that the basic Bass model is not suitable for fitting
long-run series of fast-tech product sales, such as PCs, because after the first few time periods repeats
are such a large portion of sales that the series is contaminated: it does not embody the underlying
adoption process. Our dream, and a primary motivation for this research, has been to accurately
quantify the adoption series buried in measures of sales. The dots in Figure 19 are the BB-01 output
adoptions summed over all nine PC generations. The peak in 1985 implies two adoption processes:
the character-based user interface (DOS and predecessors) process starting in 1975 and peaking in
1985 followed by the graphical-user interface (GUI, e.g., Macintosh and Windows) process starting
in 1989. The fits of both series are excellent with R-square above .95 for the first and above .99 for
FIGURE 17. U.S. PC Installed-Base Observations v. Bass-Bass Output (Left) FIGURE 18. U.S. PC Installed-Bass Generational Mix (Right)
0
20,000,000
40,000,000
60,000,000
80,000,000
100,000,000
120,000,000
140,000,000
160,000,000
1975 1980 1985 1990 1995 2000
Internet
Multimedia
Windows
32-Bits
Hard Drive
IBM PC Compatibility
Applications
Manufacturer
Kits
0.00E+00
2.00E+07
4.00E+07
6.00E+07
8.00E+07
1.00E+08
1.20E+08
1.40E+08
1.60E+08
1.80E+08
1975 1980 1985 1990 1995 2000
U.S. PC Installed Base Estimated by PC Industry Analyst
U.S. PC Installed Base Calculated by Bass-Bass
Page 26
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 26 of 29
the second with very high t-ratios for all parameter estimates. The Bass model identifies 2000 as the
peak in GUI adoptions and the two models predict a total U.S. market potential of 284 million. This
prediction is consistent with the following facts: (1) the BB-01 model identified 154 million adopters
at the end of 2000 and (2)
when generations eight and
nine are complete there will
be 189 million adopters of
generation nine and prior
products. The higher poten-
tial market predicted by the
basic Bass model is anticipat-
ing generations ten and later.
When one considers that more
than 50% of the members of
the 138 million U.S. civilian
workforce now have a computer at work, over 60% of the 106 U.S. households have at least one
computer with 30% of households owning more than one PC, the market potential forecast by the
Bass model seems quite reasonable. Another big change in user interface like GUI, change in form
factor like notebooks or some other new big thing might be just the ticket for another enormous un-
derlying wave in the PC market.
5. Summary and Conclusions
Fast-tech products are different: repeat purchasing is driven by user-perceived functionality increases.
Although there are some earlier product markets (e.g., mainframe computers) that show signs of fast-
tech behavior, there was an explosion in fast-tech categories spawned by semiconductor technology.
Starting in the mid 1970s, semiconductors gave form to microprocessors, DRAMs, and PCs. Pio-
neering fast-tech categories were soon followed by cellular telephone, DVD, Internet and many oth-
ers. Since 1974-75, the Moore’s Law regularity of exponential change in semiconductor
FIGURE 19. Basic Bass Model Fit of Total PC Adopters Calculated by BB-01 Model
0.0E+00
5.0E+06
1.0E+07
1.5E+07
2.0E+07
2.5E+07
1975 1980 1985 1990 1995 2000
PC Adoptions Calculated by Bass-Bass ModelBass Model Fitted Adoptions (DOS)Bass Model Fitted Adoptions (Windows)
Endodgenous Variable
R-Square
Adj R-Square
Bass Model 1 (DOS)
Adoptions(t) 0.961 0.953
Bass Model 2 (Windows)
Adoptions(t) 0.992 0.990
Param EstimateApprox Std
Errt Ratio
M1 2.98E+07 1.86E+06 16.0p1 0.0008 0.0003 2.3q1 0.5844 0.0476 12.3M2 2.54E+08 2.24E+07 11.4p2 0.0050 0.0004 11.5q2 0.3350 0.0217 15.4
Bass Model 1 (DOS)
Bass Model 2 (Windows)
Page 27
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 27 of 29
price/performance has been a drum beat signaling new capability in almost uncountable end-user
products and services.
The rapid rates of functionality change in fast-tech products have resulted in (and been the re-
sult of) rapid market adaptation to change. Buyers of earlier generations migrate rapidly to newer
generations and new buyers are brought into the market as higher functionality applications become
available. To capture the dynamics of the underlying market structure of adoption, repeat buying, and
usurping, a new model of generational interaction and market growth is required. We have developed
such a model and applied it to data for all developed generations of two of the most important fast-
tech products, one end-user product and one component product. In both cases the model fits the data
exceptionally well and, more importantly, the model produces estimates of the internal structure of
demand dynamics of adoption, repeating, usurpation, installed-base growth, installed-base mix evolu-
tion, and expected time-to-repeat that are highly plausible and consistent with independent estimates
by industry experts.
Page 28
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 28 of 29
References
Bass, Frank M. 1969. A new product growth model for consumer durables. Management Science. 15
(January) 215-227.
_____ 1995. Empirical generalizations and marketing science: a personal view. Marketing Science.
14 (3) Part 2 of 2 G6-G19.
Bayus, Barry L, Namwoon Kim, and Allan D. Shocker 2000. Growth models for multi-product inter-
actions: current status and new directions. In Mahajan, Vijay S., Eitan Muller, and Yoram Wind
(2000), New-Product Diffusion Models. Kluwer Academic Publishers, Boston / Dordrecht / Lon-
don. 141-164.
Blackman, A. W. 1974. The market dynamics of technological substitutions. Technological Forecast-
ing and Social Change. 6 (February) 41-63.
Danaher, Peter J., Bruce G.S Hardie and William P. Putsis. 2001. Marketing mix variables and the
diffusion of successive generations of a technological innovation. Journal of Marketing Research.
38 (November) 501-514.
Fisher, J.C. and R.H. Pry. 1971. A simple substitution model of technological change. Technological
Forecasting and Social Change. 3 (March) 75-88.
Islam, T. and N. Mead. 1997. The diffusion of successive generations of a technology: a more general
model. Technological Forecasting and Social Change. 56 (September) 49-60.
Johnson, W. and K. Bhatia. 1997. Technological substitution in mobile communications. Journal of
Business and Industrial Marketing. 12 (6) 383-399.
Kim, Namwoon, Dae Ryun Chang, and Allen D. Shocker 2000. Modeling inter-category and genera-
tional dynamics for a growing information technology industry. Management Science. 46 (4)
(April) 496-512.
Lehmann, Donald R. and Jae H. Pae. 2000. Multigeneration Innovation Diffusion: The Impact of
‘Intergeneration Time.’ Hong Kong Polytechnic University Working Paper.
Mahajan, Vijay and Eitan Muller. 1996. Timing, Diffusion and Substitution of Successive Genera-
tions of Technological Innovations: The IBM Mainframe Case. Technological Forecasting and
Social Change. 51 (February) 109-132.
Page 29
Diffusion of Technology Generations: A Model of Adoption and Repeat Sales
Working Paper – Portia Isaacson Bass and Frank M. Bass – 11/30/2001 -- page 29 of 29
_____, S. Sharma and Robert D. Buzzell 1993. Assessing the impact of competitive entry on market
expansion and incumbent sales. Journal of Marketing. 57 (July) 39-52.
Norton, John A. 1986. Growth, Diffusion and Technological Substitution in Industrial Markets: An
Examination of the Semiconductor Industry. Doctoral Dissertation. University of Texas at Dallas.
_____ and Frank M. Bass 1987. A diffusion theory model of adoption and substitution for successive
generations of high-technology products. Management Science. 33 (September) 1069-1086.
_____ and Frank M. Bass 1992. Evolution of technological generations: the law of capture. Sloan
Management Review. 33 (Winter) 66-77.
Peterka, V. 1977. Macrodynamics of technological change: market penetration of new technologies.
International Institute for Applied Systems Analysis, Laxenburg, Austria.